\(\int \frac {\text {arctanh}(a x)^4}{c x-a c x^2} \, dx\) [138]

Optimal result

Integrand size = 20, antiderivative size = 118 \[ \int \frac {\text {arctanh}(a x)^4}{c x-a c x^2} \, dx=\frac {\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}+\frac {2 \text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-a x}\right )}{c}-\frac {3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,-1+\frac {2}{1-a x}\right )}{c}+\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (4,-1+\frac {2}{1-a x}\right )}{c}-\frac {3 \operatorname {PolyLog}\left (5,-1+\frac {2}{1-a x}\right )}{2 c} \] Output:

arctanh(a*x)^4*ln(2-2/(-a*x+1))/c+2*arctanh(a*x)^3*polylog(2,-1+2/(-a*x+1) 
)/c-3*arctanh(a*x)^2*polylog(3,-1+2/(-a*x+1))/c+3*arctanh(a*x)*polylog(4,- 
1+2/(-a*x+1))/c-3/2*polylog(5,-1+2/(-a*x+1))/c
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.86 \[ \int \frac {\text {arctanh}(a x)^4}{c x-a c x^2} \, dx=\frac {\text {arctanh}(a x)^4 \log \left (1-e^{2 \text {arctanh}(a x)}\right )}{c}+\frac {2 \text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,e^{2 \text {arctanh}(a x)}\right )}{c}-\frac {3 \text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,e^{2 \text {arctanh}(a x)}\right )}{c}+\frac {3 \text {arctanh}(a x) \operatorname {PolyLog}\left (4,e^{2 \text {arctanh}(a x)}\right )}{c}-\frac {3 \operatorname {PolyLog}\left (5,e^{2 \text {arctanh}(a x)}\right )}{2 c} \] Input:

Integrate[ArcTanh[a*x]^4/(c*x - a*c*x^2),x]
 

Output:

(ArcTanh[a*x]^4*Log[1 - E^(2*ArcTanh[a*x])])/c + (2*ArcTanh[a*x]^3*PolyLog 
[2, E^(2*ArcTanh[a*x])])/c - (3*ArcTanh[a*x]^2*PolyLog[3, E^(2*ArcTanh[a*x 
])])/c + (3*ArcTanh[a*x]*PolyLog[4, E^(2*ArcTanh[a*x])])/c - (3*PolyLog[5, 
 E^(2*ArcTanh[a*x])])/(2*c)
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2026, 6494, 6620, 6624, 6624, 7164}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\text {arctanh}(a x)^4}{c x-a c x^2} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\text {arctanh}(a x)^4}{x (c-a c x)}dx\)

\(\Big \downarrow \) 6494

\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}-\frac {4 a \int \frac {\text {arctanh}(a x)^3 \log \left (2-\frac {2}{1-a x}\right )}{1-a^2 x^2}dx}{c}\)

\(\Big \downarrow \) 6620

\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}-\frac {4 a \left (\frac {3}{2} \int \frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{1-a^2 x^2}dx-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )}{c}\)

\(\Big \downarrow \) 6624

\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}-\frac {4 a \left (\frac {3}{2} \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )}{2 a}-\int \frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )}{1-a^2 x^2}dx\right )-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )}{c}\)

\(\Big \downarrow \) 6624

\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}-\frac {4 a \left (\frac {3}{2} \left (\frac {1}{2} \int \frac {\operatorname {PolyLog}\left (4,\frac {2}{1-a x}-1\right )}{1-a^2 x^2}dx+\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )}{2 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (4,\frac {2}{1-a x}-1\right )}{2 a}\right )-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )}{c}\)

\(\Big \downarrow \) 7164

\(\displaystyle \frac {\text {arctanh}(a x)^4 \log \left (2-\frac {2}{1-a x}\right )}{c}-\frac {4 a \left (\frac {3}{2} \left (\frac {\text {arctanh}(a x)^2 \operatorname {PolyLog}\left (3,\frac {2}{1-a x}-1\right )}{2 a}-\frac {\text {arctanh}(a x) \operatorname {PolyLog}\left (4,\frac {2}{1-a x}-1\right )}{2 a}+\frac {\operatorname {PolyLog}\left (5,\frac {2}{1-a x}-1\right )}{4 a}\right )-\frac {\text {arctanh}(a x)^3 \operatorname {PolyLog}\left (2,\frac {2}{1-a x}-1\right )}{2 a}\right )}{c}\)

Input:

Int[ArcTanh[a*x]^4/(c*x - a*c*x^2),x]
 

Output:

(ArcTanh[a*x]^4*Log[2 - 2/(1 - a*x)])/c - (4*a*(-1/2*(ArcTanh[a*x]^3*PolyL 
og[2, -1 + 2/(1 - a*x)])/a + (3*((ArcTanh[a*x]^2*PolyLog[3, -1 + 2/(1 - a* 
x)])/(2*a) - (ArcTanh[a*x]*PolyLog[4, -1 + 2/(1 - a*x)])/(2*a) + PolyLog[5 
, -1 + 2/(1 - a*x)]/(4*a)))/2))/c
 

Defintions of rubi rules used

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 6494

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x 
_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - 
Simp[b*c*(p/d)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))] 
/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c 
^2*d^2 - e^2, 0]
 

rule 6620

Int[(Log[u_]*((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^ 
2), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(PolyLog[2, 1 - u]/(2*c*d)) 
, x] + Simp[b*(p/2)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[2, 1 - u]/( 
d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d 
 + e, 0] && EqQ[(1 - u)^2 - (1 - 2/(1 - c*x))^2, 0]
 

rule 6624

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*PolyLog[k_, u_])/((d_) + (e_ 
.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(PolyLog[k + 1, u]/(2* 
c*d)), x] - Simp[b*(p/2)   Int[(a + b*ArcTanh[c*x])^(p - 1)*(PolyLog[k + 1, 
 u]/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, k}, x] && IGtQ[p, 0] && E 
qQ[c^2*d + e, 0] && EqQ[u^2 - (1 - 2/(1 - c*x))^2, 0]
 

rule 7164

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, 
x]}, Simp[w*PolyLog[n + 1, v], x] /;  !FalseQ[w]] /; FreeQ[n, x]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.52 (sec) , antiderivative size = 761, normalized size of antiderivative = 6.45

method	result	size
derivativedivides	\(\frac {\frac {a \operatorname {arctanh}\left (a x \right )^{4} \ln \left (a x \right )}{c}-\frac {a \operatorname {arctanh}\left (a x \right )^{4} \ln \left (a x -1\right )}{c}+\frac {4 a \left (\frac {\left (-2 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}+2 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}+i \pi \,\operatorname {csgn}\left (i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )-i \pi \,\operatorname {csgn}\left (i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}-i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}+2 i \pi +2 \ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{4}}{8}-\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{4}+\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{4}+\operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (5, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{4}+\operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (5, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{c}}{a}\)	\(761\)
default	\(\frac {\frac {a \operatorname {arctanh}\left (a x \right )^{4} \ln \left (a x \right )}{c}-\frac {a \operatorname {arctanh}\left (a x \right )^{4} \ln \left (a x -1\right )}{c}+\frac {4 a \left (\frac {\left (-2 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}+2 i \pi {\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}+i \pi \,\operatorname {csgn}\left (i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )\right ) \operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) \operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )-i \pi \,\operatorname {csgn}\left (i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )\right ) {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}-i \pi \,\operatorname {csgn}\left (\frac {i}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right ) {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{2}+i \pi {\operatorname {csgn}\left (\frac {i \left (-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}-1\right )}{-\frac {\left (a x +1\right )^{2}}{a^{2} x^{2}-1}+1}\right )}^{3}+2 i \pi +2 \ln \left (2\right )\right ) \operatorname {arctanh}\left (a x \right )^{4}}{8}-\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (\frac {\left (a x +1\right )^{2}}{-a^{2} x^{2}+1}-1\right )}{4}+\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{4}+\operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (5, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\operatorname {arctanh}\left (a x \right )^{4} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{4}+\operatorname {arctanh}\left (a x \right )^{3} \operatorname {polylog}\left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-3 \operatorname {arctanh}\left (a x \right )^{2} \operatorname {polylog}\left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 \,\operatorname {arctanh}\left (a x \right ) \operatorname {polylog}\left (4, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (5, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{c}}{a}\)	\(761\)
parts	\(\text {Expression too large to display}\)	\(1141\)

Input:

int(arctanh(a*x)^4/(-a*c*x^2+c*x),x,method=_RETURNVERBOSE)
 

Output:

1/a*(a/c*arctanh(a*x)^4*ln(a*x)-a/c*arctanh(a*x)^4*ln(a*x-1)+4*a/c*(1/8*(- 
2*I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))^2+2*I*Pi*csgn(I/(-(a*x+1)^2/(a^2 
*x^2-1)+1))^3+I*Pi*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1))*csgn(I/(-(a*x+1)^2/( 
a^2*x^2-1)+1))*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(-(a*x+1)^2/(a^2*x^2-1)+1 
))-I*Pi*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1))*csgn(I*(-(a*x+1)^2/(a^2*x^2-1)- 
1)/(-(a*x+1)^2/(a^2*x^2-1)+1))^2-I*Pi*csgn(I/(-(a*x+1)^2/(a^2*x^2-1)+1))*c 
sgn(I*(-(a*x+1)^2/(a^2*x^2-1)-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))^2+I*Pi*csgn(I 
*(-(a*x+1)^2/(a^2*x^2-1)-1)/(-(a*x+1)^2/(a^2*x^2-1)+1))^3+2*I*Pi+2*ln(2))* 
arctanh(a*x)^4-1/4*arctanh(a*x)^4*ln((a*x+1)^2/(-a^2*x^2+1)-1)+1/4*arctanh 
(a*x)^4*ln(1+(a*x+1)/(-a^2*x^2+1)^(1/2))+arctanh(a*x)^3*polylog(2,-(a*x+1) 
/(-a^2*x^2+1)^(1/2))-3*arctanh(a*x)^2*polylog(3,-(a*x+1)/(-a^2*x^2+1)^(1/2 
))+6*arctanh(a*x)*polylog(4,-(a*x+1)/(-a^2*x^2+1)^(1/2))-6*polylog(5,-(a*x 
+1)/(-a^2*x^2+1)^(1/2))+1/4*arctanh(a*x)^4*ln(1-(a*x+1)/(-a^2*x^2+1)^(1/2) 
)+arctanh(a*x)^3*polylog(2,(a*x+1)/(-a^2*x^2+1)^(1/2))-3*arctanh(a*x)^2*po 
lylog(3,(a*x+1)/(-a^2*x^2+1)^(1/2))+6*arctanh(a*x)*polylog(4,(a*x+1)/(-a^2 
*x^2+1)^(1/2))-6*polylog(5,(a*x+1)/(-a^2*x^2+1)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.31 \[ \int \frac {\text {arctanh}(a x)^4}{c x-a c x^2} \, dx=\frac {\log \left (\frac {2 \, a x}{a x - 1}\right ) \log \left (-\frac {a x + 1}{a x - 1}\right )^{4} + 4 \, {\rm Li}_2\left (-\frac {2 \, a x}{a x - 1} + 1\right ) \log \left (-\frac {a x + 1}{a x - 1}\right )^{3} - 12 \, \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} {\rm polylog}\left (3, -\frac {a x + 1}{a x - 1}\right ) + 24 \, \log \left (-\frac {a x + 1}{a x - 1}\right ) {\rm polylog}\left (4, -\frac {a x + 1}{a x - 1}\right ) - 24 \, {\rm polylog}\left (5, -\frac {a x + 1}{a x - 1}\right )}{16 \, c} \] Input:

integrate(arctanh(a*x)^4/(-a*c*x^2+c*x),x, algorithm="fricas")
 

Output:

1/16*(log(2*a*x/(a*x - 1))*log(-(a*x + 1)/(a*x - 1))^4 + 4*dilog(-2*a*x/(a 
*x - 1) + 1)*log(-(a*x + 1)/(a*x - 1))^3 - 12*log(-(a*x + 1)/(a*x - 1))^2* 
polylog(3, -(a*x + 1)/(a*x - 1)) + 24*log(-(a*x + 1)/(a*x - 1))*polylog(4, 
 -(a*x + 1)/(a*x - 1)) - 24*polylog(5, -(a*x + 1)/(a*x - 1)))/c
 

Sympy [F]

\[ \int \frac {\text {arctanh}(a x)^4}{c x-a c x^2} \, dx=- \frac {\int \frac {\operatorname {atanh}^{4}{\left (a x \right )}}{a x^{2} - x}\, dx}{c} \] Input:

integrate(atanh(a*x)**4/(-a*c*x**2+c*x),x)
 

Output:

-Integral(atanh(a*x)**4/(a*x**2 - x), x)/c
 

Maxima [F]

\[ \int \frac {\text {arctanh}(a x)^4}{c x-a c x^2} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{4}}{a c x^{2} - c x} \,d x } \] Input:

integrate(arctanh(a*x)^4/(-a*c*x^2+c*x),x, algorithm="maxima")
 

Output:

-1/80*log(-a*x + 1)^5/c + 1/16*integrate(-(log(a*x + 1)^4 - 4*log(a*x + 1) 
^3*log(-a*x + 1) + 6*log(a*x + 1)^2*log(-a*x + 1)^2 - 4*log(a*x + 1)*log(- 
a*x + 1)^3)/(a*c*x^2 - c*x), x)
 

Giac [F]

\[ \int \frac {\text {arctanh}(a x)^4}{c x-a c x^2} \, dx=\int { -\frac {\operatorname {artanh}\left (a x\right )^{4}}{a c x^{2} - c x} \,d x } \] Input:

integrate(arctanh(a*x)^4/(-a*c*x^2+c*x),x, algorithm="giac")
 

Output:

integrate(-arctanh(a*x)^4/(a*c*x^2 - c*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arctanh}(a x)^4}{c x-a c x^2} \, dx=\int \frac {{\mathrm {atanh}\left (a\,x\right )}^4}{c\,x-a\,c\,x^2} \,d x \] Input:

int(atanh(a*x)^4/(c*x - a*c*x^2),x)
 

Output:

int(atanh(a*x)^4/(c*x - a*c*x^2), x)
 

Reduce [F]

\[ \int \frac {\text {arctanh}(a x)^4}{c x-a c x^2} \, dx=\frac {\mathit {atanh} \left (a x \right )^{5}-5 \left (\int \frac {\mathit {atanh} \left (a x \right )^{4}}{a^{2} x^{3}-x}d x \right )}{5 c} \] Input:

int(atanh(a*x)^4/(-a*c*x^2+c*x),x)
 

Output:

(atanh(a*x)**5 - 5*int(atanh(a*x)**4/(a**2*x**3 - x),x))/(5*c)